Peers and peer networks are important proximal social influences across the lifespan starting at an early age, and, from a developmental perspective, few other microsystemic contexts are as important to children's healthy development. Indeed, the extent to which children form relationships with peers, and the nature and quality of those relationships, significantly influences a wide range of health-related behaviors and developmental outcomes, including substance use, addiction, and delinquency. However, extant conceptual models and methodological approaches lack the sophistication needed to allow scientists to adequately develop hypotheses about how and the degree to which peers socialize healthy and unhealthy outcomes. Thus, we propose to explore new methods of mapping young children's peer connectivity onto social profiles that are in the literature and that reflect risk for drug use and addiction in adolescence. We plan to integrate concepts, methods, and innovations arising from two traditionally disparate disciplines --- child development and theoretical mathematics --- and to apply contemporary statistical and Monte Carlo modeling and data visualization procedures to them to study the structure and organization of children's peer relationships and peer socialization processes with regard to early behavioral risk factors. Specifically, we plan to further develop and refine a new approach for studying social networks, called the Q-connectivity method. This method is derived from discrete homotopy theory (also called A-theory), which is a mathematical method of modeling complex systems and their dynamics. In our preliminary research, we have demonstrated that this method can be successfully applied to describe important features of young children's peer relationships and to explain how these features relate to children's own tendencies to exhibit externalizing behaviors (Hanish, Martin et al., in press) and to early school achievement (Hanish, Barcelo et al., in press) --- factors that predict later substance use. Our specific aims are to: (1) study the properties of individual children's peer networks, and (2) study the consistency and stability of these networks over time. The concepts and methods that are generated from these aims will be applied to the following substantive question: How do individual differences among children and their peers as well as differences in interaction qualities contribute to variations in patterns of peer selection and influence along behavioral risk dimensions? We will focus specifically on early risk behaviors that are significant risk factors for later delinquency and substance use. We focus on young children for two primary reasons: (1) because early identification of children who are at risk for negative peer influences may enhance our ability to intervene before drug use and addiction occur and (2) because intensive longitudinal peer interactional data are not available for older children and adolescents;thus, we use the available data as a model for how we might apply this to older populations. PUBLIC HEALTH RELEVANCE: Individuals'health-related behaviors occur within the context of their social interactions and relationships. We propose to study how young children's general tendencies to be engaged with versus isolated from others, as well as the pattern of social relationships and interactions that they maintain, influence behaviors and patterns of functioning that can place them at risk for later substance use, addiction, and delinquency.